# Full text of "Analytical Mechanics"

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```ADDITION AND RESOLUTION OF VECTORS           13
PROBLEMS.
1.   The resultant of two vectors which are at right angles to each other is twice the smaller of the two.    The magnitude of the smaller vector is a; find the magnitude of the greater vector.
2.   In the preceding problem find the resultant vector.
3.   Find analytically the sum of three equal vectors which point in the following directions — East, N. 30° W., and S. 30° W.
4.   In the preceding problem make use of the first graphical method. 6.  In problem 3 make use of the second graphical method.
6.   A vector which is 15 cm. long points N. 30° E.   Find its components in the following directions.
(a)  N. 30° W.         (c) W.                  (e) S. 60° E.
(b)  N. 60° E.          (d) S. 30° W.       (f) E.
7.   A vector a is in the rc?/-plane.   If 3 is added to ax and 4 to ay the direction of the vector is not changed but its magnitude becomes ax + av. Find the magnitude and direction of a.
8.   Three vectors a, b, and c lie in the icy-plane.   Find their resultants analytically, taking the magnitudes of their components from the following tables:
ax      ay             bx      by             cx        cy
(1)        0,       9,           -5,       2,             0,       10.
(2)    -3,       7,              5,       0,             6,      -8.
(3)        0,  -10,              8,       5,             3,      -2.
(4)        2,       0,           -6,       4,             0,         8.
9.   In the preceding problem make use of the second graphical method.
10.   Straight horizontal tunnels in a mine connect the points Pi, I\ PI, and 7*4, in the given order.    The length of each tunnel and the angle it makeH with the meridian are given in the following tables.   Find the lengths and directions of the tunnels which have to be dug in order to connect 7\ with Pt and J\ directly.
PJ\ = 200 feet, and makes 30° with the meridian. />2/>, = J()() foot, and makes 120° with the meridian. 1\1\ = 400 feet, and makes 300° with the meridian.
11.   Work out the preceding problem by the first graphical method.
12.   Work out problem 10 by the second graphical method.
13.   Find the direction and magnitude of the force experienced by an electrical charge of five units placed at one vertex of an equilateral triangle due to two unlike charges of 10 units each placed at the other vertices. The sides of the triangle are 2 cm.```